A Model with Costly Enforcement Jesús Fernández-Villaverde University of Pennsylvania December 25, 2012 Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 1 / 43
A Model with Costly Enforcement We keep the basic structure as before, except that now the financial market friction is the cost of enforcing contracts. Structure: 1 Borrower may decide to renege on debt. 2 If that is the case, the lender can only recover the fraction (1 θ) of the gross return R k t+1 p tq t k t where: (1 θ) R k t+1 < R t and the borrower keeps the rest, θr k t+1 p tq t k t. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 2 / 43
Costly Enforcement Model II Value of project: V t = R k t+1p t q t k t R t (q t k t n t ) Incentive constraint: V t θr k t+1p t q t k t Since the constraint must be binding: Rt+1p k t q t k t R t (q t k t n t ) = θrt+1p k t q t k t 1 q t k t = n 1 (1 θ) R t+1 k t R t p t Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 3 / 43
Costly Enforcement Model III Advantage: much easier to handle than costly state verification model. Disadvantage: no default in equilibrium, no spreads. When to use each of them? Can we move from firms to banks? Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 4 / 43
An Application I: Bank Runs 2007-2010: run on investment funds instead of classical run on banks. Suggests we may want to think again about runs on financial institutions. Calling them or not a bank is somewhat irrelevant: any institution that engages in maturity transformation. Gorton (2010) s emphasis on runs on funds during the crisis. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 5 / 43
Net Repo Funding to Banks and Broker-Dealers, $billions Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 6 / 43
An Application II: Bank Runs Kiyotaki and Gertler (2012) incorporate Diamond and Dybvig (1983) into the dynamic macro model we saw in last lecture. Main idea: maturity mismatch. To keep the presentation simple, we will get rid of nominal rigidities. Also, this will facilitate comparison with a neoclassical framework. But, first, let us review Diamond and Dybvig s model. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 7 / 43
A Review of Diamond and Dybvig: Agents Continuum of agents. Three-dates economy: 1 t = 0: each agents endowed with 1 unit of good. 2 t = 1: early consumption, with probability π 1 and utility u (c 1 ). 3 t = 2: late consumption, with probability π 1 = 1 π 2 and utility u (c 2 ). Think about the need of consumption as a liquidity shock i.i.d. for each agent. Law of large numbers. Consumption in other date does not yield utility. Expected utility of agents: π 1 u (c 1 ) + π 2 u (c 2 ) Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 8 / 43
A Review of Diamond and Dybvig: Technology Technology: 1 Storage 1 good at t is transformed into 1 good t + 1. 2 Long-term illiquid investment project 1 good at time t = 0 is transformed into R > 1 at t = 2. However, if liquidated at t = 1, we get l 1. Reasons: 1 Technological. 2 Monitoring. 3 Lemons... Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 9 / 43
A Review of Diamond and Dybvig: Effi cient Allocation Social planner: perfect risk-pooling among agents. Invest I and store 1 I in such a way that no long-term project is liquidated too early. We solve Then: Optimality condition: max π 1 u (c 1 ) + π 2 u (c 2 ) s.t. π 1 c 1 = 1 I π 2 c 2 = RI ( ) ( ) 1 I RI max π 1 u + π 2 u π 1 π 2 u ( 1 I π 1 ) ( ) RI = Ru π 2 Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 10 / 43
A Review of Diamond and Dybvig: Autarky Each agent invests I in the long-term project at t = 0 and stores 1 I. If liquidity shock at t = 1, c 1 = 1 I + li = 1 (1 l) I 1 Otherwise c 2 = RI + 1 I = 1 + (R 1) I R (at least one of the two inequalities is strict). Expected utility: π 1 u (1 (1 l) I ) + π 2 u (1 + (R 1) I ) I is always ex post ineffi cient: either too low or too high. Inferior to effi cient allocation. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 11 / 43
A Review of Diamond and Dybvig: Financial Markets I p unit of good at t = 1 can be exchanged for 1 unit at t = 2. Then and Note c 1 = pc 2. c 1 = pri + 1 I c 2 = RI + 1 I p Also, utility is increasing in I if pr > 1 and decreasing if pr < 1. Thus, in a equilibrium where I is endogenous, pr = 1. Hence, allocation is c 1 = 1 and c 2 = R. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 12 / 43
A Review of Diamond and Dybvig: Financial Markets II Expected utility: π 1 u (1) + π 2 u (R) dominates autarky, but it is still not effi cient because liquidity is not properly allocated. To see this, note that, in general u (1) = Ru (R) For instance, if u (1) > Ru (R), impatient consumers get more in the optimal allocation than in the equilibrium with financial markets (she needs to be insured against the liquidity risk better than what she can get on her own by storing all her endowment). Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 13 / 43
A Review of Diamond and Dybvig: Fractional Reserve Banking I A bank can offer a contract to depositors: (c 1, c 2 ). It must be the case that c2 > c 1 (otherwise, depositors will always cash-in at t = 1 regardless of the liquidity shock). Let us suppose that agents withdraw funds when they want to consume. Then, bank keeps reserves π 1 c1 1 π 1 c1. and invests in the long-term project Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 14 / 43
A Review of Diamond and Dybvig: Fractional Reserve Banking II Payouts: c 2 = R 1 π 1c 1 π 2 By competition, (c1, c 2 ) should satisfy: or max π 1 u (c c1 1 ) + π 2 u u (c 1 ) = Ru (c 2 ) ( R 1 π 1c 1 π 2 This is the same optimality condition than the social planner! Intuition: coalition. ) Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 15 / 43
A Review of Diamond and Dybvig: Fractional Reserve Banking II Problem: what if the depositors show up at t = 1? Bank run (self-fulling prophecy). Sequential service constraint. It is a Nash, regardless of the investors beliefs about the soundness of the portfolio of the bank. Ineffi cient allocation where the bank has to liquidate early the long-run project. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 16 / 43
A Review of Diamond and Dybvig: Solutions 1 Narrow banking Wallace (1996): 1 Pay in all events: even worse than autarky. 2 Pay if liquidation: same than autarky. 3 Securitization: same than equilibrium with financial markets. 2 Suspension of Convertibility. 3 Equity: Jacklin (1986). 4 Deposit insurance. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 17 / 43
Environment We deal now with a more general model. Households and bankers. Two goods: 1 Durable asset, capital, which does not depreciate and it is in fixed supply k = 1. 2 A nondurable good. Relative price of capital: q t. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 18 / 43
Capital Capital is hold by banks and households: k b t + k h t = k = 1 When capital kt b is hold by a bank at period t, it produces z t+1 kt b of nondurable good at period t + 1. When capital kt h is hold by a household at period t, it requires f ( kt h ) to produce z t+1 kt b of nondurable good at period t + 1. Interpretation as management cost. Assumption: ( ) ( ) α f kt h 2 k h 2 t for k h t k h ( ) t (0, 1) = αk h t kt h k h t 2 for kt h > k h t Kink in management costs allows the household to absorb all the capital in case of a banking collapse. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 19 / 43
Representative Household Preferences: E 0 t=0 β t log c h t Endowment of nondurable goods z t w h. Deposits in a bank that pay R t+1 if no bank run. If bank run, a depositor receives either the full payment or nothing, depending on the timing of the withdrawal. We assume that, ex ante, the household gives zero probability to bank run. Hence, budget constraint: ( ) ct h + d t + q t kt h + f kt h = z t w h + R t d t 1 + (q t + z t ) kt 1 h Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 20 / 43
Optimality Conditions The first-order conditions for the household are: Asset pricing kernel: 1 c t = λ t λ t = βe t λ t+1 R t+1 λ t = βe t λ t+1 R h t+1 R t+1 = q t+1 + z t+1 q t + f ( ) kt h SDF t = β λ t λ t 1 and standard non-arbitrage conditions. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 21 / 43
Expression for Consumption Define: F t = z t w h + f ( k h t ) ( ) kt h f kt h + E t λ t+1 F t+1 as the sum of the value of the endowment (z t w h ) plus the returns from holding capital (f ( kt h ) k h t f ( kt h ) ) plus the continuation value. Then, from budget constraint and optimality conditions, we get: ) ct h = (1 β) (R t d t 1 + (q t + z t ) kt 1 h + F t Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 22 / 43
Bankers I Continuum of bankers. Perfect competition. Risk neutral. 1 Survival rate σ, with expected life consumption at terminal date. 1 σ (replaced by new bankers), and Equity n t such that: q t k b t = n t + d t Initial wealth w b : take it as an exogenous endowment to simplify algebra. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 23 / 43
Bankers II Evolution of equity: n t = (q t + z t ) k b t 1 R t d t 1 with c b t = n t. Then: Note the recursive structure. V t = E t (1 σ) σ i 1 β i ct+i b i=1 = E t [(1 σ) βn t+1 + βσv t+1 ] Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 24 / 43
Moral Hazard Problem I A banker can divert a fraction θ of assets for personal use and depositors can only recover 1 θ of assets. Hence, following arguments we have already presented, the incentive constraint (IC) is: V t θq t k b t Problem of the banker max V t = E t (1 σ) σ i 1 β i ct+i b i=1 s.t. n t = (q t + z t ) k b t 1 R t d t 1 V t θq t k b t Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 25 / 43
Moral Hazard Problem II Guess: V t = ν kt k b t ν t d t Why this structure? Linear preferences+expected utility. With some simple algebra: where V t = µ t q t k b t + ν t n t µ t = ν kt q t ν t is the excess marginal value of assets over deposits. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 26 / 43
Moral Hazard Problem III Now, IC is: µ t q t k b t ν t n t θq t k b t ν t n t (θ µ t ) q t k b t Hence, IC is binding if and only if θ > µ t > 0. If IC is not binding, µ t = 0, that is, competition forces down the excess marginal value of assets over deposits to zero. When µ t > 0, there is excess return to induce the right behavior and the price of capital is low. From now on, I will assume that IC is binding. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 27 / 43
Moral Hazard Problem IV Then: q t k b t n t = ν t θ µ t = φ t where φ t is the maximum leverage ratio. Interpretation. Note linearity in bank assets. Too big to fail? Alternatives? Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 28 / 43
Value of Franchise I With some algebra, we get V t = µ t q t kt b + ν t n t = E t [(1 σ) βn t+1 + βσv t+1 ] { [ ( )] 1 σ + σ νt+1 + φ = βe t+1 µ t+1 t [( ) R b t+1 R t+1 qt kt b ] + R t+1 n t } where R b t+1 = q t+1 + z t+1 q t+1 is the realized rate of return on bank assets and is the realized excess return. R b t+1 R t+1 Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 29 / 43
Value of Franchise II Then, by matching coeffi cients µ t = βe t Ω t+1 ( R b t+1 R t+1 ) ν t = βe t Ω t+1 R t+1 where Ω t+1 = 1 σ + σ ( ) ν t+1 + φ t+1 µ ( t+1 = 1 σ + σ ν t+1 + q t+1k b t+1 n t+1 µ t+1 ) is the (probability weighted) marginal value of net bank worth at time t + 1. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 30 / 43
Aggregation Total bank net worth times leverage is equal to capital owned by banks: q t k b t = φ t n t Total net worth of banks: } n t = σ {(z t + q t ) kt 1 b R t d t 1 + (1 σ) w b Consumption of bankers: } ct b = (1 σ) {(z t + q t ) kt 1 b R t d t 1 Total output: y t = z t + z t w h + (1 σ) w b ( ) = c t + ct b + f kt h Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 31 / 43
Equilibrium Conditions I The first-order conditions of the household: 1 1 = βe t R t+1 c t c t+1 1 1 q t+1 + z t+1 = βe t c t c t+1 q t + f ( ) kt h Consumption of bankers: } ct b = (1 σ) {(z t + q t ) kt 1 b R t d t 1 Evolution of wealth of bankers: } n t = σ {(z t + q t ) kt 1 b R t d t 1 + (1 σ) w b Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 32 / 43
Equilibrium Conditions II IC: q t kt b = β E t Ω t+1 R t+1 n t θ µ t The coeffi cients of the value of the franchise: ( ) qt+1 + z t+1 µ t = βe t Ω t+1 R t+1 q t+1 ( Ω t = 1 σ + σ βe t Ω t+1 R t+1 + q tkt b ) µ n t t Market clearing: q t kt b = n t + d t kt b + kt h = 1 z t + z t w h + (1 σ) w b = c t + ct b + f ( ) kt h Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 33 / 43
Bank Run I Sequential service obligation. At the start of period t, before realization of returns, depositors run on the bank: they do not roll over their deposits. Small ε cost of not rolling over the deposit (to avoid runs without foundation). Bank liquidates its capital by selling it to households at price q t. Therefore, a run is possible if some depositors will lose their assets in a run: (q t + z t ) k b t 1 < γr t d t 1 where γ is the percentage of depositors who run. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 34 / 43
Bank Run II With some algebra or (q t + z t γr t q t 1 ) k b t 1 + γr t n t 1 < 0 R b t = q t + z t q t 1 < γr t ( 1 1 ) φ t 1 Interpretation: a bank run is possible when the realized rate of return of bank assets, Rt b, in case of a liquidation, is low in comparison with the rate of return on deposits R t and the leverage ratio, φ t 1. Rt b, R t, and φ t 1 are all equilibrium objects and, hence, they depend on the state of the economy. Calibration: regular shocks do not push the economy to the bank-run region, but large shocks do. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 35 / 43
Liquidation Price I When a run occurs, banks liquidate and no new banks enter into the economy (new potential bankers just eat their initial endowment): n t = d t = 0 and R t+1 is not defined. Alternative: slow recovery. Nothing too important for what we have to discuss today. Households get all the capital: k h t = 1. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 36 / 43
Liquidation Price II Then, after a run: c b t = (1 σ) w b c t = z t + z t w h f (1) 1 1 qt+1 = βe + z t+1 t c t c t+1 qt + αk h (note: capital holdings are heterogeneous, but we assume that fees are the same). Rearranging terms and solving forward: q t = E t ( β i i=1 c ( t z t+i αk h)) αk h c t+i Difference between illiquidity and insolvency: depends on the relation between q t and q t. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 37 / 43
An Extension Why would banks issue short-term liquid deposits instead of long-term bonds? Incorporate liquidity risk by households. After all the decisions are made, some members of the household must undertake emergency consumption c m t. They will withdraw it from deposits. With minor changes, all the equilibrium conditions go through. Jesús Fernández-Villaverde (PENN) Costly-Enforcement December 25, 2012 38 / 43
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